##### Sample Reviews and Summaries of the Listed Books with comments by our guest Lee Kraftchick (LK)

__ ____A FEW CLASSICS__

** A Mathematician’s Apology, **G. H. Hardy (1940)

__Amazon Description__**:** G. H. Hardy was one of this century’s finest mathematical thinkers, renowned among his contemporaries as a ‘real mathematician … the purest of the pure’. . . . This ‘apology’, written in 1940, offers a brilliant and engaging account of mathematics as very much more than a science; when it was first published, Graham Greene hailed it alongside Henry James’s notebooks as ‘the best account of what it was like to be a creative artist’. . . . This is a unique account of the fascination of mathematics and of one of its most compelling exponents in modern times.

** LK**: Hardy is an unapologetic Platonist when it comes to the question of whether math is invented or discovered- he insists it is discovered:

“I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our ‘creations’, are simply our notes of our observations. This view has been held, in one form or another, by many philosophers of high reputation from Plato onwards, and I shall use the language which is natural to a man who holds it.”

** How to Solve It, **G. Polya (1945)

__Amazon__**:** A must-have guide by eminent mathematician G. Polya, *How to Solve It* shows anyone in any field how to think straight. In lucid and appealing prose, Polya reveals how the mathematical method of demonstrating a proof or finding an unknown can help you attack any problem that can be reasoned out—from building a bridge to winning a game of anagrams.

*How to Solve It* includes a heuristic dictionary with dozens of entries on how to make problems more manageable—from analogy and induction to the heuristic method of starting with a goal and working backward to something you already know.

This disarmingly elementary book explains how to harness curiosity in the classroom, bring the inventive faculties of students into play, and experience the triumph of discovery. But it’s not just for the classroom. Generations of readers from all walks of life have relished Polya’s brilliantly deft instructions on stripping away irrelevancies and going straight to the heart of a problem.

** Men of Mathematics, **E

*.*T. Bell (1937, copyright renewed 1986)

__Amazon__**:*** Men of Mathematics* provides a rich account of major mathematical milestones, from the geometry of the Greeks through Newton’s calculus, and on to the laws of probability, symbolic logic, and the fourth dimension. Bell breaks down this majestic history of ideas into a series of engrossing biographies of the great mathematicians who made progress possible—and who also led intriguing, complicated, and often surprisingly entertaining lives.

Never pedantic or dense, Bell writes with clarity and simplicity to distill great mathematical concepts into their most understandable forms for the curious everyday reader. Anyone with an interest in math may learn from these rich lessons, an advanced degree or extensive research is never necessary.

** Wikipedia**:

** Men of Mathematics . . .** is a book on the history of mathematics published in 1937 by Scottish-born American mathematician and science fiction writer E. T. Bell (1883–1960). After a brief chapter on three ancient mathematicians, it covers the lives of about forty mathematicians who flourished in the seventeenth through nineteenth centuries. The book is illustrated by mathematical discussions, with emphasis on mainstream mathematics.

To keep the interest of readers, the book typically focuses on unusual or dramatic aspects of its subjects’ lives. *Men of Mathematics* has inspired many young people, including John Forbes Nash Jr., Julia Robinson, and Freeman Dyson, to become mathematicians. It is not intended as a rigorous history, and includes many anecdotal accounts.

**LK**:

Sample of mathematicians discussed:

- Archimedes(287?–212 BC)
- Descartes(1596–1650)
- Fermat(1601–1665)
- Pascal(1623–1662)
- Newton(1642–1727)
- Leibniz(1646–1716)
- Euler(1707–1783)
- Cauchy(1789–1857)
- Lobachevsky(1793–1856)
- Riemann(1826–1866)
- Poincaré(1854–1912)
- Cantor(1845–1918)

*Men of Mathematics* remains widely read to this day. It has generally received high praise, but has sometimes been criticized as inaccurate or misleading:

**E.g. Positive**: Theoretical physicist Freeman Dyson called his encounter with the book one of the decisive moments in his early career path, noting its ability to present famous mathematicians not as saints, but as flawed individuals of mixed qualities who nevertheless accomplished great mathematics.^{[8]}

**Contrary view:** The mathematics profession was “poorly served by Bell’s book:”

- …perhaps the most widely read modern book on the history of mathematics. As it is also one of the worst, it can be said to have done a considerable disservice to the profession.
^{[4]}Ivor Grattan-Guinness - There is a general impression based on the widely read book of E.T. Bell that Lagrange, in his
*Méchanique Analytique*, was the first to have connected time to space as a fourth coordinate. …However, Lagrange did not express these thoughts quite as precisely as Bell seems to imply….Thus, it is far from certain after consulting the*original*text whether or not Lagrange came close to formulating, even in his own mind, the concept credited to him by Bell.^{[5]} - Clifford Truesdell wrote: “[Bell] was admired for his science fiction and his
*Men of Mathematics*. I was shocked when, just a few years later, Walter Pitts told me the latter was nothing but a string of Hollywood scenarios; my own subsequent study of the sources has shown me that Pitts was right, and I now find the contents of that still popular book to be little more than rehashes enlivened by nasty gossip and banal or indecent fancy.^{”}

*Innumeracy: Mathematical Illiteracy and its Consequences*** ,** John Allen Paulos (1988)

**LK:** Paulos explains why the general public needs to understand basic mathematics, such as how to make educated probabilistic guesses, read polls and other statistical data, and analyze scientific studies. Easy to read and very insightful.

- Good explanations of: Probability and coincidence, why correlation does not imply causation, and other statistical issues.
- Includes a few legal cases in which statistics were misused.

**Amazon: **Why do even well-educated people understand so little about mathematics? And what are the costs of our innumeracy? John Allen Paulos, in his celebrated bestseller first published in 1988, argues that our inability to deal rationally with very large numbers and the probabilities associated with them results in misinformed governmental policies, confused personal decisions, and an increased susceptibility to pseudoscience of all kinds. Innumeracy lets us know what we’re missing, and how we can do something about it.

Sprinkling his discussion of numbers and probabilities with quirky stories and anecdotes, Paulos ranges freely over many aspects of modern life, from contested elections to sports stats, from stock scams and newspaper psychics to diet and medical claims, sex discrimination, insurance, lotteries, and drug testing.

__Also by Paulos__**: A Mathematician Reads The Newspaper, Beyond Innumeracy, **and

*Once Upon a Number*** The Mathematical Experience,** Philip

**Davis and Reuben Hirsh (1981, updated 1995)**

**Short Description:**

**LK**: An introduction to the philosophy of mathematics- what is mathematics, proof, platonism (mathematics exists independent of humans and we merely discover it), formalism (math is but a game of manipulating symbols according to manmade rules). Mathematical discoveries that changed long-held assumptions- e.g., non-euclidean geometry (Lobachevsky (infinite parallel lines) Riemannian (no parallel lines), Russell’s paradox, different kinds of infinity (Cantor’s diagonal proof, showing that the set of real numbers is “larger” than the set of integers). Nicely organized into discrete topics for further exploration.

** Amazon: **Winner of the 1983 National Book Award.

Mathematics has been a human activity for thousands of years. Yet only a few people from the vast population of users are professional mathematicians, who create, teach, foster, and apply it in a variety of situations. The authors of this book believe that it should be possible for these professional mathematicians to explain to non-professionals what they do, what they say they are doing, and why the world should support them at it. They also believe that mathematics should be taught to non-mathematics majors in such a way as to instill an appreciation of the power and beauty of mathematics. Many people from around the world have told the authors that they have done precisely that with the first edition and they have encouraged publication of this revised edition complete with exercises for helping students to demonstrate their understanding. This edition of the book should find a new generation of general readers and students who would like to know what mathematics is all about. It will prove invaluable as a course text for a general mathematics appreciation course, one in which the student can combine an appreciation for the esthetics with some satisfying and revealing applications.

The text is ideal for 1) a GE course for Liberal Arts students 2) a Capstone course for perspective teachers 3) a writing course for mathematics teachers. A wealth of customizable online course materials for the book can be obtained from Elena Anne Marchisotto (elena.marchisotto@csun.edu) upon request.

*Journey Through Genius**,* William Dunham (1990)

__Amazon__**:** “Like masterpieces of art, music, and literature, great mathematical theorems are creative milestones, works of genius destined to last forever. . . . Dunham [describes some important theorems and their proofs], plac[ing] each theorem within its historical context, and explores the very human and often turbulent life of the creator — from Archimedes, the absentminded theoretician whose absorption in his work often precluded eating or bathing, to Gerolamo Cardano, the sixteenth-century mathematician whose accomplishments flourished despite a bizarre array of misadventures, to the paranoid genius of modern times, Georg Cantor. He also provides step-by-step proofs for the theorems, each easily accessible to readers with no more than a knowledge of high school mathematics.

**LK**: Sample of proofs included:

- Euclid’s Proof of the Pythagorean Theorem (ca. 300 B.C.)
- Euclid’s proof of the Infinitude of Primes (ca. 300 B.C.)
- Cantor’s proof of “the Non-Denumerability of the Continuum” (1874)

** Also by Dunham: The Mathematical Universe: An Alphabetical Journey Through the Great Proofs, Problems, and Personalities** (1994)

__MORE RECENT MATHEMATICS BOOKS __

*Wonders of Numbers: Adventures in Mathematics, Mind, and Meaning*

Clifford A. Pickover (2002)

__Amazon__**:** Who were the five strangest mathematicians in history? What are the ten most interesting numbers? Jam-packed with thought-provoking mathematical mysteries, puzzles, and games, *Wonders of Numbers* will enchant even the most left-brained of readers.

Hosted by the quirky Dr. Googol–who resides on a remote island and occasionally collaborates with Clifford Pickover–*Wonders of Numbers* focuses on creativity and the delight of discovery. Here is a potpourri of common and unusual number theory problems of varying difficulty–each presented in brief chapters that convey to readers the essence of the problem rather than its extraneous history. Peppered throughout with illustrations that clarify the problems, *Wonders of Numbers* also includes fascinating “math gossip.” How would we use numbers to communicate with aliens? Check out Chapter 30. Did you know that there is a Numerical Obsessive-Compulsive Disorder? You’ll find it in Chapter 45.

From the beautiful formula of India’s most famous mathematician to the Leviathan number so big it makes a trillion look small, Dr. Googol’s witty and straightforward approach to numbers will entice students, educators, and scientists alike to pick up a pencil and work a problem.

** Group Theory in the Bedroom,** Brian Hayes (2008)

__Kirkus__**:** A selection of “Computing Science” columns by *American Scientist *magazine’s former editor-in-chief aimed at the numerate—or at least mathematically curious—reader.

While you don’t have to be a geek to appreciate Hayes’s lively, self-effacing style (complete with afterthoughts), it helps to understand that computer science relies on a field of math called numerical analysis and uses algorithms—rules for generating solutions to problems through an iterative process (the way you learned to do square roots in high school). The first essay explains how clockmakers developed the gears and linkages that enabled fabled medieval clocks to reach remarkable accuracy, as well as predict the day Easter would fall on. Other essays celebrate the notion of random numbers and why they are so hard to achieve. Numerical analysis also plays a role in economic models based on the kinetic theory of gases or simplified markets involving iterations of buying and selling. Hayes goes on to explain how statistics have been applied to compute which quarrels—from interpersonal to world wars—are the deadliest (surprising results here). Also, he looks at how algorithms have been developed to determine ways to divide a random series of numbers into two parts with equal sums, or nearly equal sums if the series total is odd. Gears appear again in the form of algorithms, which yield practical tables of numbers to enable engineers to make gear trains to approximate complex ratios. A couple of essays probe areas only professionals might ponder, such as computing the location of the Continental Divide or why base 3 arithmetic is better than base 10 or binary systems. But the pièce de résistance is the title essay, which explains why there is no algorithm whose repetitions would cycle through all four possible mattress positions that would assure equal wear and tear over time.

Challenging but rewarding for anyone intrigued by numbers.

** How Math Explains the World, **James Stein (2008)

__Goodreads__**:** [M]athematician Stein reveals how seemingly arcane mathematical investigations and discoveries have led to bigger, more world-shaking insights into the nature of our world. In the four main sections of the book, Stein tells the stories of the mathematical thinkers who discerned some of the most fundamental aspects of our universe. From their successes and failures, delusions, and even duels, the trajectories of their innovations—and their impact on society—are traced in this fascinating narrative. Quantum mechanics, space-time, chaos theory and the workings of complex systems, and the impossibility of a “perfect” democracy are all here. Stein’s book is both mind-bending and practical, as he explains the best way for a salesman to plan a trip, examines why any thought you could have is imbedded in the number π , and—perhaps most importantly—answers one of the modern world’s toughest why the garage can never get your car repaired on time. Friendly, entertaining, and fun, How Math Explains the World is the first book by one of California’s most popular math teachers, a veteran of both “math for poets” and Princeton’s Institute for Advanced Studies. And it’s perfect for any reader wanting to know how math makes both science and the world tick.

**LK: **Very interesting explanation of the limits on mathematical and logical analysis: Some true statements cannot be proved within a given system (Godel); some systems are inherently unpredictable- chaotic; sometimes we are asking for too much- e.g., a voting system that satisfies all, seemingly reasonable, conditions; some questions have more than one possible answer, e.g. the parallel postulate; sometimes we lack sufficient information- common in the law, so we make judgment calls that cannot be readily quantified; or what constitutes beauty in art or music, too subjective for complete logical analysis.

** Is God a Mathematician?,** Mario Livio (2009)

**LK**: Drawing on a variety of sources, the author asks the fundamental questions: (1) Does mathematics have an existence independent of the human mind? And (2) why do mathematical concepts have applicability far beyond the context in which they were developed? (Wigner’s “unreasonable effectiveness” of mathematics to explain scientific theories). Is math invented or discovered? His answer: both.

The author cites such sources as Hardy’s “*A Mathematician’s Apology,*” expressing the Platonist view, *Mathematics and the Imagination,* expressing the opposing formalist view (noting that the advent of non-Euclidean geometries undermines any notion that mathematical truths exist independent of human invention), and *The Mathematical Experience*, asserting that mathematicians hold both views depending on when they are asked, the Platonist when working on a problem, the formalist when they step back.

** Kirkus Reviews**: Why does math describe reality so well? A scientist offers tentative answers.

Livio (*The Equation that Couldn’t Be Solved: How Mathematical Genius Discovered the Language of Symmetry*, 2005, etc.), an astrophysicist at the Hubble Space Telescope Science Institute, frames his investigation with a history of math, beginning with the key question: Are mathematical truths discovered or invented? Pythagoras came down firmly on the side of discovery. His argument convinced Plato, and thus almost every ancient philosopher of note. The default assumption throughout most of history was that numbers, geometric figures and other mathematical truths are *real*. Galileo was the first to argue that scientific truth was necessarily expressed in mathematical terms. Newton’s highly accurate calculations of the gravitational force drove the point home, implying that math and physical reality were two sides of the same coin. Even probability and statistics, which seem fuzzier than the hard equations of physics, give useful answers in the world of quantum interactions. But then math began to explore realms of thought that had no obvious relation to the world as we experience it: non-Euclidean geometry, or the paradoxes of set theory and symbolic logic. The idea that math was a game invented by mathematicians rather than something inherent in reality became fashionable, perhaps even inescapable. Also, it became clear that certain undeniably useful scientific disciplines—Darwinian evolution, to name one salient example—resisted mathematical treatment. Even so, Livio shows that correspondences between mathematical discoveries and physical phenomena continued to crop up, often in abstract mathematics created without any idea of practical applications, such as Einstein’s use of non-Euclidean geometry. Knot topology, devised to explain a long-discredited model of the atom, turned out to have application to string theory. The author gives no final answer to the central question of math’s relationship to reality. There are physical phenomena that are modeled by math, he asserts, but we also understand reality with a brain wired to find mathematical relations all around it.

The conclusion falls a bit flat, but Livio’s trip through mathematical history is thoroughly enjoyable and requires no special training to follow it.

** Proofiness, The Dark Arts Of Mathematical Deception,** Charles Seiffe (2010)

** Kirkus**: A short course in how politicians, lawyers, advertisers and others use numbers to deceive.

The book starts with Sen. Joseph McCarthy’s claim that 207 communists were working in the State Department. The number changed over the following weeks, but once it was out there, people bought it—the apparent precision made it credible. The misuse of numbers and statistics is commonplace in our society, as the author demonstrates with plenty of absurd statistics that collapse under even the slightest examination. “Potemkin numbers”—those invented to sell a preconceived idea—are just one variety of abuse; the inherent inexactness of measurement yields many bogus numbers. The 98.6 degrees “normal” body temperature is an average based on readings of armpit temperature, rarely used by modern medicine. Similarly misleading are wild extrapolations from current data. One journal published data showing that female marathoners would at some future date post faster times than men. But women began to run the race only recently, so their records reflect a much smaller sample. Extrapolated further, those same numbers show that women runners will eventually break the sound barrier. Polls are especially subject to error, writes Seife, due to the very nature of sampling. The vaunted “margin of error” is widely misunderstood, and can hide inaccuracies the pollsters would rather not admit to. Even elections are subject to miscounting, especially in close contests such as the 2008 Minnesota senatorial race. Stacking the deck—for example, gerrymandering election districts—can also yield results that defy the popular will. Seife favors no party, giving examples of how all segments of the political spectrum deal in bogus numbers when it fits their agenda. While nothing is likely to stop the merchandising of misleading statistics and Potemkin numbers, readers of this book will at least have some protection when the next slick huckster tries to bamboozle them with fancy figures.

Sprightly written, despite its sobering message.

** Here’s Looking at Euclid: From Counting Ants to Games of Chance – An Awe-Inspiring Journey Through the World of Numbers,** Alex Bellos

**(2011)**

** Kirkus:** An expansive overview of numbers and figures, and those who find them irresistible.

Though he has an Oxford degree in math, former *Guardian *reporter Bellos (*Futebol: Soccer: The Brazilian Way*, 2002) approaches the subject as an enthusiastic amateur. He begins at the most basic level, with the concept of number itself, looking at the ways children, tribal cultures and animals deal with the idea of quantity. Perhaps not surprisingly, an ability to recognize which of two trees bears the most fruit seems to predate the ability to count. Cultural differences appear even in mathematically advanced societies, and the conventional system of base ten math is only one of several ways to break up the number system, with binary math probably the best known alternative. For arithmetic, Bellos looks at Japanese abacus experts, who can add columns of numbers faster than a calculator, and the Vedic math promoted by an Indian sect, which offers advanced algorithms for multiplication and other troublesome operations. Geometry also provides plenty of material, from the Pythagorean theorem to origami to the “golden ratio” beloved by architects and artists. A chapter on logarithms leads to a discussion of slide rules, the first choice for scientists and technicians requiring a quick answer until the pocket calculator drove it out of favor. Another chapter provides a lucid discussion of statistics and the famous bell curve. Recreational math gets its due, as well, with nods to Sudoku, Rubik’s Cube and the master puzzler Martin Gardner. The final chapter examines infinities and non-Euclidean geometry. Bellos maintains focus on the people who have created math and who have used it creatively, from the famous Greeks to Renaissance figures like Descartes and Fermat, and 19th-century giants like Gauss and Poincaré. Readers desiring more will find online appendices that treat the concepts more rigorously, with proofs where relevant. However, most readers who remember high-school math can follow the clear and entertaining accounts.

A smorgasbord for math fans of all abilities.

*The Joy of x: A Guided Tour of Math, from One to Infinity*** ,** Steven Strogatz (2013)

__Amazon__**: **“Delightful . . . easily digestible chapters include plenty of helpful examples and illustrations. You’ll never forget the Pythagorean theorem again!”**— Scientific American**

Many people take math in high school and promptly forget much of it. But math plays a part in all of our lives all of the time, whether we know it or not. In *The Joy of x*, Steven Strogatz expands on his hit *New York Times* series to explain the big ideas of math gently and clearly, with wit, insight, and brilliant illustrations.

Whether he is illuminating how often you should flip your mattress to get the maximum lifespan from it, explaining just how Google searches the internet, or determining how many people you should date before settling down, Strogatz shows how math connects to every aspect of life. Discussing pop culture, medicine, law, philosophy, art, and business, Strogatz is the math teacher you wish you’d had. Whether you aced integral calculus or aren’t sure what an integer is, you’ll find profound wisdom and persistent delight in *The Joy of x*.

** How Not to Be Wrong: The Power of Mathematical Thinking**, Jordan Ellenberg

**(2015)**

**Amazon: **The *Freakonomics* of math—a math-world superstar unveils the hidden beauty and logic of the world and puts its power in our hands

The math we learn in school can seem like a dull set of rules, laid down by the ancients and not to be questioned. In *How Not to Be Wrong*, Jordan Ellenberg shows us how terribly limiting this view is: Math isn’t confined to abstract incidents that never occur in real life, but rather touches everything we do—the whole world is shot through with it.

Math allows us to see the hidden structures underneath the messy and chaotic surface of our world. It’s a science of not being wrong, hammered out by centuries of hard work and argument. Armed with the tools of mathematics, we can see through to the true meaning of information we take for granted: How early should you get to the airport? What does “public opinion” really represent? Why do tall parents have shorter children? Who really won Florida in 2000? And how likely are you, really, to develop cancer?

*How Not to Be Wrong* presents the surprising revelations behind all of these questions and many more, using the mathematician’s method of analyzing life and exposing the hard-won insights of the academic community to the layman—minus the jargon. Ellenberg chases mathematical threads through a vast range of time and space, from the everyday to the cosmic, encountering, among other things, baseball, Reaganomics, daring lottery schemes, Voltaire, the replicability crisis in psychology, Italian Renaissance painting, artificial languages, the development of non-Euclidean geometry, the coming obesity apocalypse, Antonin Scalia’s views on crime and punishment, the psychology of slime molds, what Facebook can and can’t figure out about you, and the existence of God.

Ellenberg pulls from history as well as from the latest theoretical developments to provide those not trained in math with the knowledge they need. Math, as Ellenberg says, is “an atomic-powered prosthesis that you attach to your common sense, vastly multiplying its reach and strength.” With the tools of mathematics in hand, you can understand the world in a deeper, more meaningful way. *How Not to Be Wrong* will show you how.

**LK**: Includes an interesting discussion of the limits of reason. Not everything can be deduced from a set of axioms, no matter how well constructed. Legal issues, for example, may sometimes be subject to logical, algorithmic analysis, but the variety of human behavior and competing principles make it impossible to decide all such issues in a strictly formalist fashion. E.g., The 2000 election case could not be decided on strictly logical, principled grounds, due to the lack of resources, uniform standards, time constraints and the need for a prompt, definitive decision.

*A Divine Language,*** Learning Algebra, Geometry, And Calculus At The Edge Of Old Age,** Alec Wilkinson (2022)

** NYT Review: **Decades after struggling to understand math as a boy, Alec Wilkinson decides to embark on a journey to learn it as a middle-aged man. What begins as a personal challenge―and it’s challenging―soon transforms into something greater than a belabored effort to learn math. Despite his incompetence, Wilkinson encounters a universe of unexpected mysteries in his pursuit of mathematical knowledge and quickly becomes fascinated; soon, his exercise in personal growth (and torture) morphs into an intellectually expansive exploration.

In *A Divine Language*, Wilkinson, a contributor to *The New Yorker* for over forty years, journeys into the heart of the divine aspect of mathematics―its mysteries, challenges, and revelations―since antiquity. As he submits himself to the lure of deep mathematics, he takes the reader through his investigations into the subject’s big questions―number theory and the creation of numbers, the debate over math’s human or otherworldly origins, problems and equations that remain unsolved after centuries, the conundrum of prime numbers. Writing with warm humor and sharp observation as he traverses practical math’s endless frustrations and rewards, Wilkinson provides an awe-inspiring account of an adventure from a land of strange sights. Part memoir, part metaphysical travel book, and part journey in self-improvement, *A Divine Language* is one man’s second attempt at understanding the numbers in front of him, and the world beyond.

__Books by Ian Stewart__ (a previous guest), a professional mathematician and prolific, engaging writer of mathematics books. He has written some fifty books, including:

** Concepts of Modern Mathematics,** Ian Stewart (1995).

** Amazon: **Some years ago, “new math” took the country’s classrooms by storm. Based on the abstract, general style of mathematical exposition favored by research mathematicians, its goal was to teach students not just to manipulate numbers and formulas, but to grasp the underlying mathematical concepts. The result, at least at first, was a great deal of confusion among teachers, students, and parents. Since then, the negative aspects of “new math” have been eliminated and its positive elements assimilated into classroom instruction.

In this charming volume, a noted English mathematician uses humor and anecdote to illuminate the concepts underlying “new math”: groups, sets, subsets, topology, Boolean algebra, and more. According to Professor Stewart, an understanding of these concepts offers the best route to grasping the true nature of mathematics, in particular the power, beauty, and utility of *pure *mathematics. No advanced mathematical background is needed (a smattering of algebra, geometry, and trigonometry is helpful) to follow the author’s lucid and thought-provoking discussions of such topics as functions, symmetry, axiomatics, counting, topology, hyperspace, linear algebra, real analysis, probability, computers, applications of modern mathematics, and much more.

By the time readers have finished this book, they’ll have a much clearer grasp of how modern mathematicians look at figures, functions, and formulas and how a firm grasp of the ideas underlying “new math” leads toward a genuine comprehension of the nature of mathematics itself.

** Do Dice Play God?** (2019)

** What’s the Use? How Mathematics Shapes Everyday Life **(2021)

__ESSAY COLLECTIONS__

*The World Of Mathematics**:*** A Small Library Of The Literature Of Mathematics From A’h-Mose The Scribe To Albert Einstein **(4 volumes), James Newman ed. (1956), as reprinted in 1988)

**LK**: Essays on a wide variety of mathematical topics, from counting and arithmetic to artificial intelligence, written by such mathematical luminaries as Isaac Newton, Rene DesCartes, Bertrand Russell, Leonard Euler, Henri Poincare, G. H. Hardy, Alan Turing and many more. All accessible to the general reader. It’s a little dated, perhaps, but much of the mathematics described is timeless.* *

** What Is Mathematics?, **Richard Courant & Herbert Robbins, as revised by Ian Stewart (2d Ed. 1996)

__LK__**:** Introduction to advanced mathematics, covering basic number theory, analysis, algebra, geometry and topology.

__Amazon__**:** For more than two thousand years a familiarity with mathematics has been regarded as an indispensable part of the intellectual equipment of every cultured person. Today, unfortunately, the traditional place of mathematics in education is in grave danger. The teaching and learning of mathematics has degenerated into the realm of rote memorization, the outcome of which leads to satisfactory formal ability but does not lead to real understanding or to greater intellectual independence. This new edition of Richard Courant’s and Herbert Robbins’s classic work seeks to address this problem. Its goal is to put the *meaning* back into mathematics.

Written for beginners and scholars, for students and teachers, for philosophers and engineers, *What is Mathematics?, Second Edition* is a sparkling collection of mathematical gems that offers an entertaining and accessible portrait of the mathematical world. Covering everything from natural numbers and the number system to geometrical constructions and projective geometry, from topology and calculus to matters of principle and the Continuum Hypothesis, this fascinating survey allows readers to delve into mathematics as an organic whole rather than an empty drill in problem solving. With chapters largely independent of one another and sections that lead upward from basic to more advanced discussions, readers can easily pick and choose areas of particular interest without impairing their understanding of subsequent parts.

Brought up to date with a new chapter by Ian Stewart, *What is Mathematics?, Second Edition* offers new insights into recent mathematical developments and describes proofs of the Four-Color Theorem and Fermat’s Last Theorem, problems that were still open when Courant and Robbins wrote this masterpiece, but ones that have since been solved.

Formal mathematics is like spelling and grammar–a matter of the correct application of local rules. Meaningful mathematics is like journalism–it tells an interesting story. But unlike some journalism, the story has to be true. The best mathematics is like literature–it brings a story to life before your eyes and involves you in it, intellectually and emotionally. *What is Mathematics* is like a fine piece of literature–it opens a window onto the world of mathematics for anyone interested to view.

** The Princeton Companion to Mathematics **(2008)

** Goodreads: **This is a one-of-a-kind reference for anyone with a serious interest in mathematics. Edited by Timothy Gowers, a recipient of the Fields Medal, it presents nearly two hundred entries, written especially for this book by some of the world’s leading mathematicians, that introduce basic mathematical tools and vocabulary; trace the development of modern mathematics; explain essential terms and concepts; examine core ideas in major areas of mathematics; describe the achievements of scores of famous mathematicians; explore the impact of mathematics on other disciplines such as biology, finance, and much, much more.

Unparalleled in its depth of coverage, *The Princeton Companion to Mathematics* surveys the most active and exciting branches of pure mathematics. Accessible in style, this is an indispensable resource for undergraduate and graduate students in mathematics as well as for researchers and scholars seeking to understand areas outside their specialties.

– Features nearly 200 entries, organized thematically and written by an international team of distinguished contributors

– Presents major ideas and branches of pure mathematics in a clear, accessible style

– Defines and explains important mathematical concepts, methods, theorems, and open problems

– Introduces the language of mathematics and the goals of mathematical research

– Covers number theory, algebra, analysis, geometry, logic, probability, and more

– Traces the history and development of modern mathematics

– Profiles more than ninety-five mathematicians who influenced those working today

– Explores the influence of mathematics on other disciplines

– Includes bibliographies, cross-references, and a comprehensive index

*The **Colossal** Book of Mathematics*** , **Martin Gardner (2001)

__Amazon__**: **No amateur or math authority can be without this ultimate compendium from America’s best-loved mathematical expert.

Whether discussing hexaflexagons or number theory, Klein bottles or the essence of “nothing,” Martin Gardner has single-handedly created the field of “recreational mathematics.” The Colossal Book of Mathematics collects together Gardner’s most popular pieces from his legendary “Mathematical Games” column, which ran in Scientific American for twenty-five years. Gardner’s array of absorbing puzzles and mind-twisting paradoxes opens mathematics up to the world at large, inspiring people to see past numbers and formulas and… more

** The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics,**Clifford Pickover (2011)

**Amazon: **The Neumann Prize–winning, illustrated exploration of mathematics—from its timeless mysteries to its history of mind-boggling discoveries.

Beginning millions of years ago with ancient “ant odometers” and moving through time to our modern-day quest for new dimensions, *The Math Book* covers 250 milestones in mathematical history. Among the numerous delights readers will learn about as they dip into this inviting anthology: cicada-generated prime numbers, magic squares from centuries ago, the discovery of pi and calculus, and the butterfly effect. Each topic is lavishly illustrated with colorful art, along with formulas and concepts, fascinating facts about scientists’ lives, and real-world applications of the theorems.

** New York Times Book of Mathematics **(2013)

__Amazon__**: **“Some of the pieces included here are important and some are curiosities, but all are absorbing . . . Recommended for casual and serious math enthusiasts.” —*Library Journal*

From the archives of the world’s most famous newspaper comes a collection of its very best writing on mathematics. Big and informative, *The New York Times Book of Mathematics* gathers more than 110 articles written from 1892 to 2010 that cover statistics, coincidences, chaos theory, famous problems, cryptography, computers, and many other topics. Edited by Pulitzer Prize finalist and senior Times writer Gina Kolata.

__A GOOD SELF- TEACHING TEXT FOR THOSE INTERESTED IN EXPLORING MATHEMATICS IN MORE DETAIL __

** Mathematics for the Nonmathematician, **Morris Kline (1985)

__Amazon__**: **Practical, scientific, philosophical, and artistic problems have caused men to investigate mathematics. But there is one other motive which is as strong as any of these — the search for beauty. Mathematics is an art, and as such affords the pleasures which all the arts afford.” In this erudite, entertaining college-level text, Morris Kline, Professor Emeritus of Mathematics at New York University, provides the liberal arts student with a detailed treatment of mathematics in a cultural and historical context. The book can also act as a self-study vehicle for advanced high school students and laymen.

Professor Kline begins with an overview, tracing the development of mathematics to the ancient Greeks, and following its evolution through the Middle Ages and the Renaissance to the present day. Subsequent chapters focus on specific subject areas, such as “Logic and Mathematics,” “Number: The Fundamental Concept,” “Parametric Equations and Curvilinear Motion,” “The Differential Calculus,” and “The Theory of Probability.” Each of these sections offers a step-by-step explanation of concepts and then tests the student’s understanding with exercises and problems. At the same time, these concepts are linked to pure and applied science, engineering, philosophy, the social sciences or even the arts.

In one section, Professor Kline discusses non-Euclidean geometry, ranking it with evolution as one of the “two concepts which have most profoundly revolutionized our intellectual development since the nineteenth century.” His lucid treatment of this difficult subject starts in the 1800s with the pioneering work of Gauss, Lobachevsky, Bolyai and Riemann, and moves forward to the theory of relativity, explaining the mathematical, scientific and philosophical aspects of this pivotal breakthrough. *Mathematics for the Nonmathematician* exemplifies Morris Kline’s rare ability to simplify complex subjects for the nonspecialist.

__SPECIFIC TOPICS__

__Geometry__

** Euclid’s Window**:

**Leonard Mlodinow (2002)**

*The Story of Geometry from Parallel Lines to Hyperspace,*__Goodreads__**: **Through *Euclid’s Window* Leonard Mlodinow brilliantly and delightfully leads us on a journey through five revolutions in geometry, from the Greek concept of parallel lines to the latest notions of hyperspace. Here is an altogether new, refreshing, alternative history of math revealing how simple questions anyone might ask about space — in the living room or in some other galaxy — have been the hidden engine of the highest achievements in science and technology.

Based on Mlodinow’s extensive historical research; his studies alongside colleagues such as Richard Feynman and Kip Thorne; and interviews with leading physicists and mathematicians such as Murray Gell-Mann, Edward Witten, and Brian Greene, *Euclid’s Window* is an extraordinary blend of rigorous, authoritative investigation and accessible, good-humored storytelling that makes a stunningly original argument asserting the primacy of geometry. For those who have looked through *Euclid’s Window,* no space, no thing, and no time will ever be quite the same.

__Calculus__

** A Tour of the Calculus, **David Berlinski (1995)

__Goodreads__**:** Were it not for the calculus, mathematicians would have no way to describe the acceleration of a motorcycle or the effect of gravity on thrown balls and distant planets, or to prove that a man could cross a room and eventually touch the opposite wall. Just how calculus makes these things possible and in doing so finds a correspondence between real numbers and the real world is the subject of this dazzling book by a writer of extraordinary clarity and stylistic brio. Even as he initiates us into the mysteries of real numbers, functions, and limits, Berlinski explores the furthest implications of his subject, revealing how the calculus reconciles the precision of numbers with the fluidity of the changing universe.

__Statistics__

** The Signal and the Noise: Why So Many Predictions Fail – But Some Don’t,** Nate Silver (2012)

** The Drunkard’s Walk: How Randomness Rules Our Lives,** Leonard Mlodinow (2009)

__Godel’s Incompleteness theorems__

** Incompleteness, The Proof and Paradox of Kurt Godel,** Rebecca Goldstein (2006)

*Gödel: A Life of Logic***, **John L. Casti and Werner DePauli (**2001)**

** Mathematics: The Loss Of Certainty,** Morris Kline (1980)

** When Einstein Walked with Gödel: Excursions to the Edge of Thought, **Jim Holt (2018)

*Gödel’s Proof*** , **Ernest Nagel, James R. Newman (1931), edited, with a new foreword by Douglas R. Hofstadter (2001)

__Amazon__**: **An accessible explanation of Kurt Gödel’s groundbreaking work in mathematical logic.

In 1931 Kurt Gödel published his fundamental paper, “On Formally Undecidable Propositions of Principia Mathematica and Related Systems.” This revolutionary paper challenged certain basic assumptions underlying much research in mathematics and logic. Gödel received public recognition of his work in 1951 when he was awarded the first Albert Einstein Award for achievement in the natural sciences―perhaps the highest award of its kind in the United States. The award committee described his work in mathematical logic as “one of the greatest contributions to the sciences in recent times.”

However, few mathematicians of the time were equipped to understand the young scholar’s complex proof. Ernest Nagel and James Newman provide a readable and accessible explanation to both scholars and non-specialists of the main ideas and broad implications of Gödel’s discovery. It offers every educated person with a taste for logic and philosophy the chance to understand a previously difficult and inaccessible subject.

__Other Topics__

** Zero: The Biography of a Dangerous Idea, **Charles Seife (2000)

__Amazon__**:** Popular math at its most entertaining and enlightening. “*Zero *is really something”-Washington Post

A New York Times Notable Book.

The Babylonians invented it, the Greeks banned it, the Hindus worshiped it, and the Church used it to fend off heretics. Now it threatens the foundations of modern physics. For centuries the power of zero savored of the demonic; once harnessed, it became the most important tool in mathematics. For zero, infinity’s twin, is not like other numbers. It is both nothing and everything.

In ** Zero**, Science Journalist Charles Seife follows this innocent-looking number from its birth as an Eastern philosophical concept to its struggle for acceptance in Europe, its rise and transcendence in the West, and its ever-present threat to modern physics. Here are the legendary thinkers—from Pythagoras to Newton to Heisenberg, from the Kabalists to today’s astrophysicists—who have tried to understand it and whose clashes shook the foundations of philosophy, science, mathematics, and religion. Zero has pitted East against West and faith against reason, and its intransigence persists in the dark core of a black hole and the brilliant flash of the Big Bang. Today, zero lies at the heart of one of the biggest scientific controversies of all time: the quest for a theory of everything.

** e: The Story of a Number, **Eli Maor (2011)

** Amazon:** The interest earned on a bank account, the arrangement of seeds in a sunflower, and the shape of the Gateway Arch in St. Louis are all intimately connected with the mysterious number

*e*. In this informal and engaging history, Eli Maor portrays the curious characters and the elegant mathematics that lie behind the number. Designed for a reader with only a modest mathematical background, this biography brings out the central importance of

*e*to mathematics and illuminates a golden era in the age of science.

** An Imaginary Tale, **Paul Nahin (2010)

** Amazon:** Today complex numbers have such widespread practical use–from electrical engineering to aeronautics–that few people would expect the story behind their derivation to be filled with adventure and enigma. In

*An Imaginary Tale*, Paul Nahin tells the 2000-year-old history of one of mathematics’ most elusive numbers, the square root of minus one, also known as

*i*. He recreates the baffling mathematical problems that conjured it up, and the colorful characters who tried to solve them.

In 1878, when two brothers stole a mathematical papyrus from the ancient Egyptian burial site in the Valley of Kings, they led scholars to the earliest known occurrence of the square root of a negative number. The papyrus offered a specific numerical example of how to calculate the volume of a truncated square pyramid, which implied the need for *i*. In the first century, the mathematician-engineer Heron of Alexandria encountered *I *in a separate project, but fudged the arithmetic; medieval mathematicians stumbled upon the concept while grappling with the meaning of negative numbers, but dismissed their square roots as nonsense. By the time of Descartes, a theoretical use for these elusive square roots–now called “imaginary numbers”–was suspected, but efforts to solve them led to intense, bitter debates. The notorious *i* finally won acceptance and was put to use in complex analysis and theoretical physics in Napoleonic times.

Addressing readers with both a general and scholarly interest in mathematics, Nahin weaves into this narrative entertaining historical facts and mathematical discussions, including the application of complex numbers and functions to important problems, such as Kepler’s laws of planetary motion and ac electrical circuits. This book can be read as an engaging history, almost a biography, of one of the most evasive and pervasive “numbers” in all of mathematics.

** A History of Pi,** Petr Beckmann (2015)

__Amazon__**: **The history of pi, says the author, though a small part of the history of mathematics, is nevertheless a mirror of the history of man. Petr Beckmann holds up this mirror, giving the background of the times when pi made progress — and also when it did not, because science was being stifled by militarism or religious fanaticism.

** A Most Elegant Equation,** David Stipp (2017)

** Book Jacket description:** An award-winning science writer introduces us to mathematics using the extraordinary equation that unites five of mathematics’ most important numbers

Bertrand Russell wrote that mathematics can exalt “as surely as poetry.” This is especially true of one equation: ei(pi) + 1 = 0, the brainchild of Leonhard Euler, the Mozart of mathematics. More than two centuries after Euler’s death, it is still regarded as a conceptual diamond of unsurpassed beauty. Called Euler’s identity or God’s equation, it includes just five numbers but represents an astonishing revelation of hidden connections. It ties together everything from basic arithmetic to compound interest, the circumference of a circle, trigonometry, calculus, and even infinity. In David Stipp’s hands, Euler’s identity formula becomes a contemplative stroll through the glories of mathematics. The result is an ode to this magical field.

__NOVELS WITH MATHEMATICAL THEMES__

__An all-time classic:__

*Flatland**, *A. Square / Edwin A. Abbot (1884)** **

**LK:** Imagine how life would be in less than three dimensions.

__Wikipedia__**:**

** Flatland: A Romance of Many Dimensions** is a satirical novella by the English schoolmaster Edwin Abbott Abbott, first published in 1884 by Seeley & Co. of London. Written pseudonymously by “A Square”, the book used the fictional two-dimensional world of Flatland to comment on the hierarchy of Victorian culture, but the novella’s more enduring contribution is its examination of dimensions

** The Curious Incident of the Dog in the Night-Time,** Mark Haddon (2004)

**LK:** A gifted boy tries to solve a mystery using mathematical techniques.

*A Certain Ambiguity, A Mathematical Novel***, **Guarav Suri & Hartosh Singh Bal (2007)

While taking a class on infinity at Stanford in the late 1980s, Ravi Kapoor discovers that he is confronting the same mathematical and philosophical dilemmas that his mathematician grandfather had faced many decades earlier — and that had landed him in jail. Charged under an obscure blasphemy law in a small New Jersey town in 1919, Vijay Sahni is challenged by a skeptical judge to defend his belief that the certainty of mathematics can be extended to all human knowledge — including religion. Together, the two men discover the power — and the fallibility — of what has long been considered the pinnacle of human certainty, Euclidean geometry.

As grandfather and grandson struggle with the question of whether there can ever be absolute certainty in mathematics or life, they are forced to reconsider their fundamental beliefs and choices. Their stories hinge on their explorations of parallel developments in the study of geometry and infinity — and the mathematics throughout is as rigorous and fascinating as the narrative and characters are compelling and complex.

Moving and enlightening, *A Certain Ambiguity* is a story about what it means to face the extent — and the limits — of human knowledge.

Winner of the 2007 Award for Best Professional/Scholarly Book in Mathematics, Association of American Publishers

** Uncle Petros and Goldbach’s Conjecture, A Novel of Mathematical Obsesion, **Apostolos Doxiadis (2010)

__Goodreads__**: **

In this critically acclaimed international bestseller, Petros Papachristos, a mathematical prodigy, has devoted much of his life trying to prove one of the greatest mathematical challenges of all Goldbach’s Conjecture, the deceptively simple claim that every even number greater than two is the sum of two primes. His feverish and singular pursuit of this goal has come to define his life. Now an old man, he is looked on with suspicion and shame by his family-until his ambitious young nephew intervenes.

Seeking to understand his uncle’s mysterious mind, the narrator of this novel unravels his story, a dramatic tale set against a tableau of brilliant historical figures-among them G. H. Hardy, the self-taught Indian genius Srinivasa Ramanujan, and a young Kurt Gödel. Meanwhile, as Petros recounts his own life’s work, a bond is formed between uncle and nephew, pulling each one deeper into mathematical obsession, and risking both of their sanity.

** Mr. Penumbra’s 24-Hour Bookstore: A Novel, **Robin Sloan (2013)

__Amazon__**: **A Winner of the Alex Award, a finalist for the *Los Angeles Times* Book Prize for First Fiction, named a Best Book of the Year by NPR, *Los Angeles Times*, and *San Francisco Chronicle*

The Great Recession has shuffled Clay Jannon away from life as a San Francisco web-design drone and into the aisles of Mr. Penumbra’s 24-Hour Bookstore. But after a few days on the job, Clay discovers that the store is more curious than either its name or its gnomic owner might suggest. The customers are few, and they never seem to buy anything―instead, they “check out” large, obscure volumes from strange corners of the store. Suspicious, Clay engineers an analysis of the clientele’s behavior, seeking help from his variously talented friends. But when they bring their findings to Mr. Penumbra, they discover the bookstore’s secrets extend far beyond its walls. Rendered with irresistible brio and dazzling intelligence, Robin Sloan’s *Mr. Penumbra’s 24-Hour Bookstore* is exactly what it sounds like: an establishment you have to enter and will never want to leave.** **

__AND A DIFFICULT TO CATEGORIZE__**, BUT BRILLIANT, BOOK, WEAVING TOGETHER FICTION, MUSIC, PHILOSOPHY, ART AND SERIOUS MATHEMATICS:**

*Gödel, Escher, Bach** An **Eternal** Golden Braid**,* Douglas Hofstadter (1979)

** Top 5 Math Books of All Time: **Unlocking the Wonders of Mathematics, Przemek Chojecki: “This Pulitzer Prize-winning book explores the intersection of mathematics, art, and music. Hofstadter delves into the minds of mathematician Kurt Gödel, artist M.C. Escher, and composer Johann Sebastian Bach to reveal the hidden connections between their works. This book is a thought-provoking journey that challenges the reader’s understanding of logic, symmetry, and intelligence.”

As Hofstadter, himself, explains in a preface for the “Twentieth-anniversary Edition”:

(…) let me try one last time to say why I wrote this book, what it is about, and what it’s principal thesis is.

In a word, GEB is a very personal attempt to say how it is that animate beings can come out of inanimate matter. What is self, and how can a self come out of stuff that is as selfless as a stone or a puddle? What is an “I”, and why are such things found (at least so far) only in association with, as poet Russel Edson once wonderfully phrased it, “teetering bulbs of dread and dream” – that is, only in association with certain kinds of gooey lumps encased in hard protective shells mounted atop mobile pedestals that roam the world on pairs of slightly fuzzy, jointed stilts? (…)

It is heavily related to Artificial Inteligence and what, in his opinion, causes sentience to emerge. He relates it to what he calls a “strange loop”, which are a special kind of loops that can emerge in a complex interconnected system of possibly inanimate parts.

** Amazon:** Twenty years after it topped the bestseller charts, Douglas R. Hofstadter’s

*Gödel, Escher, Bach: An Eternal Golden Braid*is still something of a marvel. Besides being a profound and entertaining meditation on human thought and creativity, this book looks at the surprising points of contact between the music of Bach, the artwork of Escher, and the mathematics of Gödel. It also looks at the prospects for computers and artificial intelligence (AI) for mimicking human thought. For the general reader and the computer techie alike, this book still sets a standard for thinking about the future of computers and their relation to the way we think.

Hofstadter’s great achievement in *Gödel, Escher, Bach* was making abstruse mathematical topics (like undecidability, recursion, and ‘strange loops’) accessible and remarkably entertaining. Borrowing a page from Lewis Carroll (who might well have been a fan of this book), each chapter presents dialogue between the Tortoise and Achilles, as well as other characters who dramatize concepts discussed later in more detail. Allusions to Bach’s music (centering on his *Musical Offering*) and Escher’s continually paradoxical artwork are plentiful here. This more approachable material lets the author delve into serious number theory (concentrating on the ramifications of Gödel’s Theorem of Incompleteness) while stopping along the way to ponder the work of a host of other mathematicians, artists, and thinkers.

The world has moved on since 1979, of course. The book predicted that computers probably won’t ever beat humans in chess, though Deep Blue beat Garry Kasparov in 1997. . . . Yet *Gödel, Escher, Bach* remains a remarkable achievement. Its intellectual range and ability to let us visualize difficult mathematical concepts help make it one of this century’s best for anyone who’s interested in computers and their potential for *real* intelligence. *–Richard Dragan*

**Also by Hofstadter: **

** I Am a Strange Loop** (2007).

The book “examin[es] in depth the concept of a *strange loop* to explain the sense of “I”. The concept of a *strange loop* was originally developed in his 1979 book *Gödel, Escher, Bach*.

In the end, we are self-perceiving, self-inventing, locked-in mirages that are little miracles of self-reference.

— *Douglas Hofstadter, I Am a Strange Loop, p. 363*

__Wikipedia__**:**

Hofstadter had previously expressed disappointment with how *Gödel, Escher, Bach*, which won the 1980 Pulitzer Prize for general nonfiction, was received. In the preface to its 20th anniversary edition, Hofstadter laments that the book was perceived as a hodgepodge of neat things with no central theme. He states: “GEB is a very personal attempt to say how it is that animate beings can come out of inanimate matter. What is a self, and how can a self come out of stuff that is as selfless as a stone or a puddle?”^{[1]}

Hofstadter seeks to remedy this problem in *I Am a Strange Loop* by focusing on and expounding the central message of *Gödel, Escher, Bach*. He demonstrates how the properties of self-referential systems, demonstrated most famously in Gödel’s incompleteness theorems, can be used to describe the unique properties of minds.^{[2]}

As an exploration of the sense of “I”, Hofstadter explores his own life and those to whom he has been close.

The book received favorable reviews. The *Wall Street Journal* referred to the book as “fascinating”, “original”, and “thought-provoking”.

** Metamagical Themas: Questing for the Essence of Mind and Pattern** (1985)

** Wikipedia**: An eclectic collection of articles that Douglas Hofstadter wrote for the popular science magazine

*Scientific American*during the early 1980s. The anthology was published in 1985 by Basic Books.

The volume is substantial in size and contains extensive notes concerning responses to the articles and other information relevant to their content. (One of the notes—page 65—suggested memetics for the study of memes.)

Major themes include: self-reference in memes, language, art and logic; discussions of philosophical issues important in cognitive science/AI; analogies and what makes something similar to something else (specifically what makes, for example, an uppercase letter ‘A’ recognizable as such); and lengthy discussions of the work of Robert Axelrod on the prisoner’s dilemma, as well as the idea of superrationality.

The concept of superrationality, and its relevance to the Cold War, environmental issues and such, is accompanied by notes on experiments conducted by the author at the time. Another notable feature is the inclusion of two dialogues in the style of those appearing in *Gödel, Escher, Bach*. Ambigrams are mentioned.

There are three articles centered on the Lisp programming language, in which Hofstadter first details the language itself, and then shows how it relates to Gödel’s incompleteness theorem. Two articles are devoted to Rubik’s Cube and similar puzzles. Many chapters open with an illustration of an extremely abstract alphabet, yet one which is still gestaltly recognizable as such.